In the diagram, is a right angle. Point is on , and bisects . Points and are on and , respectively, so that and . Given that and , find the integer closest to the area of quadrilateral .Source: NCTM Mathematics Teacher

SOLUTIONAngle Bisector TheoremThe angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle.

Divided Triangle Proposition
When a triangle is divided into two smaller triangles by a segment from one of its vertices to the opposite side (whether or not this segment is an angle bisector), the ratio of the areas of the two smaller triangles is equal to the ratio of the triangles’ bases.
By the Pythagorean theorem

Calculate the area of is the angle bisector of of . By the Angle Bisector theorem
By the Divided Triangle propositionCalculate the area of is the angle bisector of of . By the Angle Bisector theorem
By the Divided Triangle proposition